An algorithmic characterization and spectral analysis of canonical splitting signed graph ξ(Σ)

An ordered pair Σ=(Σu,σ) is called the signed graph, where Σu=(V,E) is an underlying graph and σ is a signed mapping, called signature, from E to the sign set {+,−}. A marking of Σ is a function μ:V(Σ)→{+,−}. The canonical marking of a signed graph Σ, denoted μσ, is given as μσ(v)=Πvu∈E(Σ)σ(vu). The canonical splitting signed graphξ(Σ) of a signed graph Σ is defined as a signed graph ξ(Σ)=(V(ξ),E(ξ)) , with V(ξ)=V(Σ)∪V′, where V′ is copy of a vertex set in V(Σ) s.t. for each vertex u∈V(Σ), take a new vertex u′ and E(ξ) is defined as, join u′ to all the vertices of Σ adjacent to u by negative edge if μσ(u)=μσ(v)=−, where v∈N(u) and by positive edge otherwise. The objective of this paper is to propose an algorithm for the generation of a canonical splitting signed graph, a splitting root signed graph from a given signed graph, provided it exists and to give the characterization of balanced canonical splitting signed graph. Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and Laplacian matrices of the canonical splitting signed graph to study its eigenvalues and eigenvectors. A relationship between the energy of the original signed graph Σ and the energy of the canonical splitting signed graph ξ(Σ) is established. • Algorithm to generate canonical splitting signed graph ξ(Σ).• Spectral Analysis is performed for both adjacency and Laplacian matrices of canonical splitting signed graph ξ(Σ).


Introduction
The initial notation and terminology used in this paper have been sourced from Harary [ 1 ], Zaslavsky [ 2 ] and West [ 3 ].The graphs examined in this paper are finite and simple.A signed graph, Σ = (Σ  , ) , is composed of an underlying graph, Σ  = ( ,  ) , where | | =  & || =  , and a signature,  ∶  → {+ , − } , which labels each edge of Σ as either ' + ' or ' − '.In this paper, edges labeled with ' + ' are considered positive and are depicted using solid lines, while edges labeled with ' − ' are considered negative and are depicted using dashed lines.If all edges in Σ are signed ' + ' or ' − ', the signed graph is referred to as homogeneous, otherwise, it is heterogeneous.Graphs can be thought of as homogeneous signed graphs with each edge being labeled as ' + '.A cycle in a signed graph Σ is considered positive if it includes an even number of negative edges.If every cycle in Σ is positive, then Σ is defined as a balanced signed graph.
An ordered pair (Σ, ) is known as a marked signed graph where Σ = (Σ  , σ) is a signed graph, and  ∶  (Σ  ) → {+ , − } , is a function defined on the vertex set  (Σ  ) of Σ  .The function  assigns each vertex of Σ  to either the positive or negative sign from the set {+ , − } , and is called the marking of Σ.
The adjacency matrix of Σ, whose vertices are  1 ,  The spectrum of a matrix is a list of its eigenvalues along with their multiplicities.Since,  (Σ) is a symmetric matrix with real entries so all its eigenvalues are real.Let  1 (Σ) >  2 (Σ) > ... >   (Σ) are distinct eigenvalues of  (Σ) along with their multiplicities  1 ,  2 , ...,   , 1 ≤  ≤  , then the list of eigenvalues of adjacency matrix is called adjacency spectrum of the signed graph Σ and usually denoted as: Let (Σ) = [ ,  ] be a diagonal matrix of order  such that the entry (,  ) is deg ( u i ) if  =  and 0 otherwise, where deg ( u i ) denotes the degree of vertex u i .(Σ) is called degree matrix of the signed graph Σ.The Laplacian matrix  (Σ) = [ ,  ] of a signed graph Σ is a square matrix of order  such that  ,  =  ,  −  ,  for 0 ≤ ,  ≤  .The eigenvalues of the Laplacian matrix is called Laplacian spectrum and is denoted by  (Σ) .Two signed graphs are said to be co-spectral if they have same spectrum.The largest eigenvalue  1 (Σ) is called the index of Σ, whereas the largest absolute eigenvalue is called spectral radius (Σ) , i.e.
The study of graph spectrum is of significant importance in the field of graph theory, and spectral graph-theoretic techniques have been applied in a range of fields including quantum physics, chemistry, computer science, and more.
In recent years, researchers have explored the spectral properties of graphs constructed through graph operations such as disjoint union, Cartesian product, Kronecker product, strong product, lexicographic product, corona, edge corona, and neighbourhood corona.A comprehensive overview of results on the spectra of these graphs can be found in the literature [4][5][6][7][8][9][10][11][12] and one can find the properties of derived signed graphs in [13][14][15][16][17][18].In [ 19 ] authors presented the idea of the splitting graph Γ(Σ  ) for a given graph Σ  .The process of creating the splitting graph Γ(Σ  ) involves taking a new vertex  ′ for each vertex  in graph Σ  .The new vertex  ′ is then connected to all vertices in Σ  that are adjacent to  .The resulting graph is referred to as the splitting graph Γ(Σ  ) of graph Σ  .
Recently, a variation of this concept has been applied in the analysis of online social networks (OSNs), where  ′ is also connected to  .This variation of the concept is referred to as the " " of  .For the purpose of convenience, the term " " is adopted for  ′ in the splitting graph Γ(Σ  ) as well.By leveraging the signed properties of splitting signed graphs, it becomes possible to develop an alternative approach to encryption and decryption, which can be employed to enhance network security.
Gutman [ 20 ] introduced the concept of energy of a graph Σ  in 1978 as the sum of the absolute values of its eigenvalues, denoted by (Σ  ) , i.e.,

𝐸( Σ
Later, in 2004, Bapat et al. [ 21 ] proved that the energy of a graph can only be an even integer if it is a rational number.Pirzada et al. [ 22 ], on the other hand, demonstrated that the energy of a given graph can never be the square root of an odd integer.Graph energy is briefly discussed in [ 8 ], while in [ 23 ] authors established a relationship between the energy of a graph and its splitting graph.The concept of graph energy has been widely studied in graph theory and has significant applications in various fields.The study of graph energy can provide insights into the structural properties of the graph and is often used in the design and analysis of communication networks, molecular chemistry, and social networks.Additionally, the energy of a graph is closely related to its spectrum and can be used to investigate various graph invariants, such as chromatic number, clique number, and independence number.Sinha et al. [ 19 ] introduced the splitting signed graphs as an extension of the splitting graph concept.The splitting signed graph of a signed graph Σ = ( , , ) , denoted as Γ(Σ) = ( Γ ,  Γ ,  Γ ) , is obtained by creating a new vertex  ′ for each vertex  ∈  (Σ) , and connecting  ′ to all vertices in Σ adjacent to  such that the sign of the corresponding edges is preserved, i.e.,  Γ ( ′  ) = ( ) for all  ∈ (  ) .In [ 24 ] authors defined, the canonical splitting signed graph (Σ) of a signed graph Σ is defined as a signed graph (Σ) = ( ( ) , ( ) ) , with  ( ) =  (Σ)  ′ , where  ′ is copy of a vertex set in  (Σ) s.t. for each vertex  ∈  (Σ) , take a new vertex  ′ and ( ) is defined as, join  ′ to all the vertices of Σ adjacent to  by negative edge if   (  ) =   (  ) = − , where  ∈ (  ) and by positive edge otherwise.Throughout we write the canonical splitting signed graph as  −     ℎ .This construction is depicted in Fig. 1 .A signed graph Σ is called a splitting signed graph if it is isomorphic to the splitting signed graph Γ(  )((  ) ) of some signed graph  , where  is referred to as the splitting root signed graph of Σ.
In this research paper, we give an algorithm to generate ( Σ) and its variant, the splitting root signed graph, from a given signed graph.We also perform spectral analysis on the adjacency and Laplacian matrices of the -splitting signed graph to investigate its eigenvalues and eigenvectors.Furthermore, we establish a relationship between the energy of the original signed graph and the energy of the -splitting signed graph.Our study sheds light on the properties of the -splitting signed graph, and its potential applications in various domains.
The Kronecker product (or tensor product) of matrices  and  is a matrix defined as follows: where,  ∈   × and  ∈   × .
Theorem 1 [ 25 ] .Let  and  are two square matrices such that  ∈   and  ∈   .If   is an eigenvalue of  with its corresponding eigenvector   , and   is an eigenvalue of  with its corresponding eigenvector   , then     is an eigenvalue of the Kronecker product  ⊗  , with the corresponding eigenvector   ⊗   .
The  − split t ing signed graph concept, the algorithm for its generation, its spectral analysis and energy based comparison become an important area to be explored.These contributions can advance the field of signed graph theory and graph analysis.
In this paper, we begin by presenting a numerical interpretation for obtaining a  − split t ing signed graph , followed by an algorithm along with its complexity analysis.Subsequently, we provide structural characterizations for identifying the splitting root signed graph, along with an accompanying algorithm and its computational complexity.Next, we establish a relationship between the spectrum of a signed graph and its  − split t ing signed graph .Finally, we investigate the energy relation.

Numerical interpretation to obtain -splitting signed graph
The procedure for generating a splitting graph can be described as follows: Given a graph with  vertices, the first step is to encode an  ×  symmetric adjacency matrix for the graph.Since a new vertex is created for each vertex in the original graph, the splitting graph will have a total of 2  vertices.The non-zero entries in the first row of the adjacency matrix indicate the vertices which are adjacent to the first vertex i.e.  1 .These entries in first row are also considered adjacent to vertex   +1 and are updated in the output matrix.
This process is repeated for each row until all rows have been processed.As a result, a 2  × 2  output matrix is generated.The adjacency matrices of the original signed graph Σ and its -splitting signed graph (Σ) can be represented as follows: It has been observed that the adjacency matrix, of order 2  , of the  −     ℎ (Σ) can be partitioned into four matrices each of order  , where the initial matrix  1 =  (Σ) , the remaining two non zero matrices are equal, say  1 , and the fourth is null matrix.Surprisingly, where Additionally, it has been discovered that the original signed graph Σ is an induced subgraph of its −     ℎ (Σ) .

Computational complexity
Computational complexity analysis is an essential aspect of evaluating the performance of an algorithm.In this regard, we analyze the complexity involved in Steps 3 and 4 of our algorithm.In these steps, we traverse each vertex of the signed graph and examine its adjacency with all other vertices.As a result, the complexity involved in these steps is ( 2 ) .Again, we analyze the complexity involved in Steps 9 and 10 of our algorithm.In these steps, we traverse each vertex of the signed graph and examine its adjacency with all other vertices.As a result, the complexity involved in these steps is ( 2 ) .
Therefore, the complexity of the proposed algorithm for finding a signed split graph with a given adjacency matrix is ( 2 ) + ( 2 ) = ( 2 ) , where  represents the number of vertices in the signed graph.

Structural characterization to derive splitting root signed graph
Sampathkumar and Walikar [ 26 ] presented the characterization of splitting graphs, which is utilized to derive the splitting root graph.
Here, we present a structural characterization of a  −     ℎ that can be used to derive the splitting root signed graph.
Theorem 2. Let Σ be a connected signed graph, then Σ is -splitting signed graph if and only if the following two conditions hold: (a) The underlying graph Σ  is splitting graph (b) ( ′ ) = − , whenever   (  ) and   (  ) both are negative in the induced graph on vertex set  1 of Σ.

Proof. Necessity
Let Σ be a connected -splitting signed graph of a signed graph Σ 1 , then it can be established that the above conditions hold for Σ.This is because Σ = (Σ 1 ) , so Σ  = (Σ  1 ) , which satisfies condition (a).Now, during the construction of Σ from Σ 1 , a new vertex  ′ is added for each vertex  in Σ 1 .These new vertices  ′ are unique to each vertex  and are stored in  2 .Meanwhile, the original vertices from Σ 1 are stored in  1 .Based on the definition of (Σ 1 ) , it follows that for each vertex  that is adjacent to  and is contained in

Deriving splitting root signed graph through structural characterization
The process for deriving the splitting root signed graph involves several steps.Firstly, it is necessary for the number of vertices, denoted as " ", to be even in order for a splitting root signed graph to exist.If " " is odd, then it is impossible to construct a splitting root signed graph.Once it has been established that " " is even, an  dimensional matrix is generated to represent the given signed graph.The matrix is examined to count the number of negative and positive edges, and it is determined whether both counts are divisible by 3. If this condition is met, a splitting root signed graph can be created; otherwise, it cannot.
The next step involves calculating the number of negative and positive edges for every vertex by counting the number of −1  and 1  in each row.If it is possible to partition the vertex set  (Σ) into two sets, such that the number of negative and positive edges in one set is exactly double of the other set, then a splitting root signed graph can be constructed.
If a splitting root signed graph exists, it's adjacency matrix of order  can be partitioned into four matrices of equal order  2 .Let [ , ] , [ , ] , [ , ] and [ , ] are the four matrices of order  2 , then  , =  , =  , for 0 ≤ ,  ≤  2 and  , = 0 for 0 ≤ ,  ≤  2 i.e. [ , ] is null matrix.To construct the matrix same and null, it is necessary to interchange rows and columns.This will result in a matrix of size  2 ×  2 .An example of the computation of a splitting root signed graph from a given signed graph will be provided in the following section.
The adjacency matrix of this signed graph,  (Σ 1 ) is a 8 × 8 matrix.Here the number of vertices in a signed graph is even, so we can find its splitting root signed graph.However, if the vertices are odd in numbers, then splitting root signed graph of the given signed graph will not exist.In this case, there are 6 occurrences of the value − 1 and 18 occurrences of the value 1 in  (Σ 1 ) .Clearly, here both the values are multiple of 3, so the splitting root signed graph can be computed.
The next step is to count the number of 1  and −1  in each row of the matrix: With this knowledge, the vertex set is splitted into two sets,  1 and  2 , where the number of 1  and −1  in  1 is double that of in  2 .In this example, 1 is {1 , 3 , 5 , 7 } and  2 is {2 , 4 , 6 , 8 } .If the partition is successful, the splitting root signed graph exists, otherwise it does not.
In this example, the 8 × 8 input matrix is divided into four equal matrices of size 4 × 4 .The fourth matrix constructed as null matrix and the first, second, and third matrices are made identical through transformations in rows and columns.The output matrix is 4 × 4 .
After applying transformations such that  2 ⇔  7 and  2 ⇔  7 , we get: and then  4 ⇔  5 and  4 ⇔  5 , we get: The first, second and third matrices can be seen to be the same, and the fourth matrix is zero, which satisfies the requirements for a splitting root signed graph.The final output matrix is the splitting root signed graph: Vertex   1 7 3 5 The signed graph  1 and its corresponding splitting root signed graph  are depicted in the Fig. 2 .

Computational complexity
The algorithm calculates the total number of positive and negative edges and traverses every entry of the matrix.As a result, the complexity to count the edges is ( 2 ) .
In Steps 30 to 52, the complexity is ( 2 ) .If the function is denoted by fun then the complexity required for the column and row transformations from Step 67 to Step 77 to construct all three matrices equal is ( 3 ) .If the fourth submatrix is already zero in the steps 78 to 91 then we use row operation to construct all other sub matrices equal.We traverse each vertex of the signed graph and examine every entry if it is equal.As a result, the complexity involved in these steps is ( 2 ×  ×  ) = ( 4 ) .Therefore, the complexity of the proposed algorithm for finding a root signed split graph with a given adjacency matrix is  ( 2 ) +  ( 3 ) + ( 4 ) +  ( 2 ) =  ( 4 ) , where  represents the number of vertices in the signed graph.

Characterization of balanced C-splitting signed graph
From Eq. ( 2) , can see that spectral radius of Σ will always be greater than equal to the index, i.e.

𝜌( Σ) ≥ 𝜆 1 ( Σ)
In [ 4 ] Acharya provided the spectral criterion for balance in Σ as, Theorem 4. A signed graph Σ is balanced if and only if it is cospectral to it's underlying graph.
It provides that the balanced signed graphs have the spectral radius equal to the index.So, here in this section we characterize the balanced  −     ℎ .
Sampathkumar provided an important characterization of balanced signed graphs based on marking: Theorem 5.The balance of a signed graph Σ = (Σ  , σ) can be determined if and only if there is a marking  of its vertices such that the sign of each edge  in Σ satisfies the condition ( ) = (  ) (  ) The operation of changing the sign of every edge in a signed graph Σ to its opposite, based on the marking  of its vertices, is called switching Σ with respect to .This operation is performed whenever the end vertices of an edge have opposite signs in Σ  .The concept of switching signed graphs is closely connected to the concept of balance, as indicated by the following theorem: Theorem 6.A signed graph Σ = (Σ  , σ) is considered balanced if and only if it is equivalent under switching to its underlying graph Σ  .Theorem 7. The  −     ℎ (Σ) is balanced if and only if the signed graph Σ is balanced.
Proof.Necessity: Let us consider (Σ) is balanced.As we know that Σ is a subsigned graph of (Σ) .This gives the result Σ is balanced.Sufficiency: Assuming Σ is balanced, it follows from Theorem 5 that there is a marking  of its vertices such that for each edge  in Σ, ( ) = (  ) (  ) .Additionally, by Theorem 6 Σ can be transformed into an all positive signed graph Σ  .We can then extend  to the vertex set  (Σ) by setting ( ′ ) = (  ) and switch (Σ)  (Σ)  ′ , resulting in (Σ)  = (Σ) .

Algorithm 2
Algorithm to derive the splitting root signed graph of a given signed graph.
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Spectrum of  -splitting signed graph
In this section, we aim to find out the spectrum of the Adjacency matrix and Laplacian matrix of a  −     ℎ (Σ) when (Σ) ≅ Γ(Σ) .
Let  (Σ) be the adjacency matrix of the signed graph Σ on  vertices, and is given as follow Let  ′  be the vertex corresponding to   , 1 ≤  ≤  , which is added in Σ to construct (Σ) , such that  ( ′  ) =  (  ) , for 1 ≤  ≤  .Then the adjacency matrix of (Σ) ,  ((Σ) ) , can be expressed as a block matrix with blocks as follows We can write above adjacency matrix as, From here we can see that adjacency matrix  ((Σ) ) is a Kronecker product of the matrices  and  (Σ) , where  = ] .
Easily we can see that

}
are the eigenvalues of the matrix .
So the adjacency spectrum of the  −     ℎ (Σ) is given as 2   , ] .

Energy of  -splitting signed graph
In this section, we explore the connection between the energy of a signed graph Σ and its  − split t ing signed graph (Σ) when (Σ) ∼ = Γ(Σ) .Proof.Let  =  1 ,  2 , ...,   be the vertex set of the signed graph Σ.Then adjacency matrix of Σ is given by, Let  ′  be the vertex corresponding to   , 1 ≤  ≤  , which is added in Σ to construct (Σ) , such that ( ′  ) = (  ) , for 1 ≤  ≤  .Then the adjacency matrix of (Σ) ,  ((Σ) ) , can be expressed as a block matrix with blocks as follows We can write it as, Let  1 (Σ) ,  2 (Σ) , ...,   (Σ) are the eigenvalues of the signed graph Σ and we can observe that the eigenvalues of

Theorem 3 ( 1
1 , ( ′  ) = − whenever   (  ) =   (  ) = − ,  ∈ (  ) , and ( ′  ) = + otherwise, satisfying condition (b).Sufficiency: If conditions (a) and (b) are satisfied for a signed graph Σ, then we can create a sub signed graph Σ 1 of Σ by including only the vertices of  1 .If we find the -splitting signed graph of Σ 1 , then we find that Σ is equivalent to (Σ 1 ) , which leads to the conclusion of the theorem.[ 24 ] ) .For any signed graph Σ, (Σ) ≅ Γ(Σ) if and only if the signed graph Σ satisfies any one of the following condition: Algorithm Algorithm to derive the  −     ℎ (Σ) of a signed graph Σ.(a) ( ) = + , whenever  and  are (b) ( ) = − , whenever  and  are adjacent and (  ) = odd for every  ; (c) Σ is heterogeneous in which both the incident vertices of each negative(positive) edges are (are not) negative.

Fig. 2 .
Fig. 2. Signed graph S 1 and S is its splitting root signed graph.

1 ≤
≤ .Now consider  (Σ) be the Laplacian matrix of the signed graph Σ and  1 ,  2 , ...,   are the eigenvalues of the Laplacian matrix  (Σ).By some easy calculations we can find the Laplacian matrix  ( (Σ) ) of the  −     ℎ is a Kronecker product of the matrices  and  (Σ), where  =